No, you were correct...the Lord of the Rings DOES have six parts. There are six books. two per novel. The story is broken up into six distinct books. Really, all the divisions are sort of superficial, as Tolkien always intended for it to be released as one part. the three novel division was sorta arbitrary, although the splitting of the story into 6 different books was always there.

I know I'm splitting hairs unbelievably, since there weren't SIX MOVIES of the Lord of the Rings, but I'm just trying to give you an out